Arnob Alan Elias Honors Level Questions: 1) As part of her new gardening delirium, Andrea wants to make something to place all her gardening pots in a very special area. She wants to carve out a square marble tile with an area of 8 + 2 15. She uses a precision cutter in whichthe length being cut must be in the form 𝑎 + 𝑏. After cutting out her tile, she notices shecan cut out another rectangular segment of the 1 marble using double her original tile’s side length as the width, and 3 of the rectangular segment’s width as the length. What is the area of this new segment? 2) In her early days, Principal Hoyle was an avid chemistry teacher and loved performing some of her own experiments. During one boring day in the lab, she decides to mix Bohemic Acid, Rudolphomycin, and Nitrofurantoin. After extensive help from the math department, she came up with the following formula to calculate the remaining volume of her concoction after all 4 5 reactions have occurred: 2𝑦 − 16 - 1 (ml),where y = moles of Bohemic Acid. a. If she observed 7 ml of solution to remain, how many moles of Bohemic Acid did she use? b. The Chemistry Department requests 150ml of Principal Hoyle’s concoction as part of a class lab. How many moles of Bohemic Acid will Principal Hoyle need in order to fulfill the request? 3) Bronx Science has experienced heavy budget cuts due to recent changes in the government’s educational funding. As a result, Mr. Yang unexpectedly loses his job, and will have to resort to alternative methods of earning an income. On a chilly Friday night, he walks into the casino and turns to gambling in hopes of earning quick cash. He plays a high risk game 3 3 4 2 where the winnings are rigged by the following formula: 216𝑦 𝑥 𝑧 3 3 4 2 + 8𝑦 𝑥 𝑧 , where y = dollars bet (in thousands), x = number of times played, and z = number of people in the casino a. Rewrite the formula as a single expression b. If Mr. Yang were to deposit $1200, with 100 people in the casino, and play the game 15 times, what are his “estimated” winnings? 4) After several unfortunate losses, Mr. Yang, being a clever former math teacher, decides to derive a complex formula to calculate a “risk-factor” based on his gambling patterns. This formula computes a risk multiplier, where a lower value represents a safer future bet, and vice versa. His derived formula is as follows: 3 3 −4 3 2 −3 8𝑥 𝑦 • (4𝑥 𝑦) 3 −5 −2 −2 • (2𝑥 𝑦 ) , where x = number of risky bets he placed, and lost from and where y = number of safe, strategic bets he won from a. Simply Mr. Yang’s risk multiplier, in terms of x and y. b. Based on your simplification from Part A, how do Mr. Yang’s past actions and experience affect his proposed risk level? 5) Alan is a famous and prosperous architect, and has been personally selected by the prime minister of Dubai to construct an extremely lavish and modern skyscraper window shaped as a rhombus. To maximize aesthetic appeal and public support, the prime minister orders Alan to 4 ensure that d1 = 6 and d2= 2 80, where d1and d2 represent the diagonals of the rhombus. Alan has a super fancy, automatic glass cutter that can only cut in lengths formatted in 𝑎 + 𝑏. Considering Alan will cut out the window from a large piece of glass, what measurements does Alan need to input in his glass cutter? 6) NASA is engineering their brand new rocket ship which they will use to discover and research exoplanets. They have designed a new type of rocket fuel that travels at a lightning fast speed of 2 6 + 2 2 light years per hour. In orderto reach the exoplanet of K2-2016-BLG-0005-Lb, they need to travel a total of 5 3 + 2 light years. Assuming they have a sufficient amount of fuel, for how long will the astronauts need to travel to reach K2-2016-BLG-0005-Lb?Simplify as necessary 7) Jacob is an enthusiastic software engineer that was hired by Google to help produce their new futuristic artificial intelligence software. He is currently writing an algorithm that will enable 6 the AI to read words at a lighting fast speed of 8 million words per second. He wants to run a stress-test on the AI, in which it will read words from a digital copy of Lord of The Rings, 4 containing 25million words. When Jacob runsthe stress test, how many seconds will it take the AI to read all the words? 8) The number 512 can be rewritten to be expressedin the format𝑎 𝑏. Using the values ofa andb, compute (2ab)(b 𝑎) 9) Rewrite (2 3 − 3 2)2 ( 4 3 + 2 2) suchthat there are only two radical signs. Regents Level Questions: 10) Harry walks to and from school each day, on the same path, walking the same distance. One day, he decides to track how far he walks. In the morning on the way to school, he uses − 𝑥 + 1 kilometer(s). In the afternoon on the way home, one app that tells him he walked 3𝑥 he uses another app that tells him he walked 1 kilometer. What is the value of x? 11) Rewrite the following expressions into radical form: 3 a. (5x)4 3 𝑦 b. 24 1 - c. (8n3) 3 Regents Level Questions: 9 −54𝑥 𝑦 12. What does ( ) ⅔ equal? 4 13. Simplify 𝑥⁷ 𝑦² ⁰𝑧¹¹ 4 6 14. Simplify the expression 72𝑥 𝑦 3 15. Which expression is equal to 𝑎 ? 2 a. 𝑎5 3 b. 𝑎2 2 c. 𝑎3 5 d. 𝑎2 16. Rationalize the denominator: 15 7 17. A rectangle has a length that is 18 metersand a width that is 8 a. Find the area of the rectangle insimplest radicalform b. Find the perimeter insimplest radical form c. If the rectangle is enlarged such that each side is multiplied by 2, what is the new area insimplest radical form? 18. Rationalize the denominator & simplify: 6 2+ 3 19. Solve and check for extraneous solutions: 2𝑥 − 5 + 7 =10 20. Convert the following expressions into radical form: − 3 a. 𝑎 4 5 b. (2𝑞) 2 Honors Level Question: 21. Alan loves mixing chemicals in his lab at home. He created a formula to find the remaining volume(in milliliters) of his special mixture after reactions: Final Volume =( 5𝑠 − 250) − 5, wheresis the moles of Solar Solution used. a. If he measures 15 ml of solution left, how many moles of Solar Solution did he use? b. If his friend Thomas needs 100 ml of the mixture, how many moles of Solar Solution is needed? Regents + Standard Questions 22. Express in terms of radicals and positive exponents. 4 6 a. 24𝑥 𝑦 2 7 16𝑥 𝑦 6 − b. 3 49𝑥 𝑦 2 3 4 −5 21𝑥 𝑦 3 8 − 9 c. 𝑥 𝑦 ( −7 3 4 𝑥 𝑦 ) 4 5 4 23.4(3𝑚 + 7) 5 + 4 = 36 Round to the hundredths place 24. Simplify 3 2 a. 49𝑥 𝑦 b. 81𝑥 𝑦 6 7 25. Convert to Radical Form 3 a. (7y)4 - 5 b. (11z) 3 Honors Level Questions (difficulty may still vary) 4 6 26. Isosceles Trapezoid ABCD has a bottom base length of 15 𝑥 , a top base length of 7 𝑥 , 3 and a height,𝐻, of 12 8𝑥 . a. Find the area of the trapezoid b. Find the length of leg CD c. The point of intersection between 𝐻 and the bottombase is E. Find the area of B𝐻D . 27. If Vanessa does not pass her AP Physics 1 exam, her parents will be very disappointed in her and she will be kicked out of the house. To avoid being homeless in the near future, Vanessa must study for the exam. Vanessa is7 8 miles away from the local library. She cannot afford public transit fare so she must jog the entire way. Her jogging pace is8 2 miles per hour. Due to working several jobs and internships to prepare for college, Vanessa can allot a maximum 5 hours for both transit to the library, studying and transit back home. How long can Vanessa study at the library given these time constraints? Give your answer in hours. 28. Vanessa is at risk of getting cut from the school track team due to not staying in shape over the summer. To try and mediate this issue, Vanessa practices at the local running track. This running area is a perfect circle, with an area of25 5 meters squared. Vanessa wants to find the sector of the track that is not covered by tree shade, since she forgot to bring sunscreen. If 30% of the total area is not covered by tree shade, find the arc length of the section of track not covered by shade. 29. Cornelius has a natural tendency of doing something he’s not supposed to be doing. Because of this, he always fools around in Mr. Yang’s class, and gets Mr.Yang very, very angry. Mr. Yang usually gets fed up of Cornelius’ nonsense, and as a result, Cornelius gets yelled at by Mr. Yang anaverageof 20 times per class period.Determine how many class periods it will take for Cornelius to be yelled at 100 times. 30. Genafyr has two identical clay pots she built in her free time. Each of these pots have a volume of 16 cubic inches. She places both pots outside for an hour, but in different places. She places one pot in the backyard, and another near her window. Genafyr went to take a nap, but 2ℎ the sun came up as soon as she went to sleep and started melting the pots at a rate of: 16 2ℎ where h = hours. She brought back the pot near her window after 2 hours, but forgot about the pot in the backyard, which she brought back after 3 hours. She decides to recycle her melted pots, and combine them for alternative usage. Express the total volume of her recycled material in radical form. 31. Rationalize and express in simplest form: 72 8 18+15 6 Answer Key: 1) Side of the square = 8 + 60. Plug into denestingformula where a=8, b=60 a = 8+ 64−60 8− 64−60 , b = → 2 2 a = 5,b = 3 → 5 + 3 = side length of square Width of new rectangular segment = 2( 5 + 3) = 2 5 + 2 3 1 2 2 Length of new rectangular segment = 3 (2 5 + 2 3) = 3 5 + 3 3 2 2 20 4 32 8 Area of new segment = L• W = (2 5 + 2 3) (3 5 + 3 3) = 3 + 2(3 15) + 4 = 3 + 3 15 2) 5 4 5 4 5 a. 2𝑦 − 16 - 1 = 7 → 2𝑦 − 16 = 8 → (raise each side to the fourth power) → 2y - 16 = 4096 y = 65541 moles of Bohemic Acid 32 b. 2𝑦 − 156 - 1 = 150 → 2𝑦 − 156 = 151 → (raise each side to the fourth power) → 2y - 4 4 5 16 = 519885601 → y = 3 3 3 8318169621 molesof Bohemic Acid 32 3 4 3 2 3 2 3 3 4 2 3 2 3) a. 216 = 6, 𝑦 = y, 𝑥 = x 𝑥, 𝑧 = 𝑧 , therefore 216𝑦 𝑥 𝑧 = 6xy 𝑥𝑧 3 3 3 3 3 4 3 2 2 3 3 3 4 2 2 8 = 2, 𝑦 = y, 𝑥 = x 𝑥, 𝑧 = 𝑧 , therefore 8𝑦 𝑥 𝑧 = 2xy 𝑥𝑧 3 3 2 3 2 2 6xy 𝑥𝑧 + 2xy 𝑥𝑧 = 8xy 𝑥𝑧 3 2 3 b. Substituting y = 1.2, x = 15, and z = 100, 8(15)(1.2)• 15 • 100 = 1440• 150 dollars in estimated winnings 3 3 −4 3 2 −3 4) a. 8𝑥 𝑦 • (4𝑥 𝑦) 3 3 −5 −2 −2 • (2𝑥 𝑦 ) 3 3 −4 3 −3 −6 −3 → distribute exponents → 8𝑥 𝑦 • 4 𝑥 𝑦 • −2 10 4 (2 𝑥 𝑦 ) make exponents positive → 2 common factors → 3 8𝑥 3 4 𝑦 1 6 3 64𝑥 𝑦 • 3 10 4 𝑥 𝑦 4 → 3 𝑦• 𝑦•2• 4 3 2𝑥 3 𝑦• 𝑦 • 2 3 𝑥 • 𝑥𝑦 3 • 3 3 → the 𝑦 ‘s cancels out → 1 2 4𝑥 𝑦 • 3 𝑥 𝑦• 𝑥𝑦 3 4 → cancel out 3 𝑥 • 𝑥 3 2𝑦• 4 b. Our simplification tells us the more risky bets he places and loses from, his risk multiplier will be higher, therefore resulting in a riskier future bet. Likewise, the more safe, strategic bets he wins from, his risk multiplier will be lower, therefore resulting in a safer future bet 5) *B y properties of rhombuses, the diagonals are perpendicularbisectors of each other 6) Let x = hours 2 6 + 2 2• x = (5 3 + 2) → x = 5 3+ 2 2 6+ 2 2 • 2 6− 2 2 2 6− 2 2 = 5 3+ 2 2 6+ 2 2 10 18−1 0 6+ 2 12− 4 16 → 5 18− 5 6+ 1 12− 2 → simplify further → 8 15 2− 5 6+2 3− 2 seconds to reach K2-2016-BLG-0005-Lb. 8 4 7) 25 6 8 2 → 1 (5 ) 4 3 (2 ) 1 6 1 → 2 5 1 2 2 → 5 2 • 2 2 10 = 2 4 8) The prime factorization of 512 is 2• 2 • 2• 2• 2• 2• 2• 2• 2, so2 willbe on the outside, 2 will be on the inside → 16 2 → a = 16, b = 2 (substitute it) → (2(16)(2))(2 16) → (64)(8) → 512 9) (2 3 − 3 2)2 =(2 3 − 3 2)• (2 3 − 3 2) = 12 - 2(2 3) (3 2) + 18 = 30 - 12 6 (30 - 12 6)(4 3 + 2 2) = 120 3 + 60 2 - 48 18 - 24 12 → simplify further → 120 3 + 60 2 - 144 2 - 48 3 → combine like-radicals → 72 3 - 84 2 − 𝑥 + 1 = 1 → 3𝑥 = 1 + 𝑥 + 1 →square both sides → 3x = 1 + 2 𝑥 + 1 + x + 1 → 10) 3𝑥 2x = 2 + 2 𝑥 + 1 → x -1 = 𝑥 + 1 → square both sides→ x2 - 2x + 1 = x + 1 → x2 - 3x = 0 → x(x-3)=0 → x = 0, x= 3 → plug it back into original equation → x = 3 (reject x = 0) 𝑥 𝑦 𝑦 𝑥 11) Key to these questions is to remember𝑎 = 𝑎 = ( 𝑎) x 4 3 a. 125𝑥 𝑦 b. 13824 c. 1 3 ( ) → 1 3 8𝑛 1 3 8𝑛 1 → 2𝑛 𝑦 12-21) 24 3 49 7 22a. Simplify the coefficients first 16 = 2 = 4 6 2 3𝑥 𝑦 3𝑥 2𝑥 𝑦 2𝑦 = 2 7 22b. Simplify the coefficients first 21 = 3 6 − 3 49𝑥 𝑦 2 −5 3 4 = 11 7𝑥 5 3𝑦 2 21𝑥 𝑦 22c. Simplify the inside 8 − 3 𝑥 𝑦 9 ( −7 3 4 4 ) 5 = ( 𝑥 𝑦 15 𝑥 3 1 12 4 ) 5 = 𝑦 12 𝑥 2 5 60 = 𝑦 12 𝑥 13 15 𝑦 4 5 23.4(3𝑚 + 7) + 4 = 36 Simplify subtract 4 then divide 4 from both sides of the equation 4 (3𝑚 + 7) 5 = 8 5 Raise both sides by the power of 4 to get ridof the exponent 5 3𝑚 + 7 = 8 4 Subtract 7 from both sides 5 4 3𝑚 = 8 − 7 Divide 3 from both sides 5 𝑚 = 4 −7 8 ≈2.15 3 24. Simplify a. 3 2 49𝑥 𝑦 Expand first 7 • 7 • 𝑥 • 𝑥 • 𝑥 • 𝑦 • 𝑦 Move the pairs out 7𝑥𝑦 𝑥 6 7 b. 81𝑥 𝑦 Recognize three pairs of x’s and 3 pairs of y’s 3 3 9𝑥 𝑦 𝑦 25. 4 a. 343𝑦 b. 1 3 5 161051𝑧 26. 4 1 6 3 a. 2 (15 𝑥 + 7 𝑥 )• 12 8𝑥 1 1 (15 2 Area = 2 (𝑏1 + 𝑏2)( ℎ) = 4 4 6 3 15 7 3 15 7 𝑥 + 7 𝑥 )• 12 8𝑥 = 2 x2 + 2 x3 • 12 8𝑥 = 2 x2 + 2 x3 • 15 24x 2𝑥 =84𝑥 + 2 x2 • 2𝑥 b. To find the length of leg CD, we must use the Pythagorean Theorem. Measures of the legs can be found using the height of the trapezoid and difference in the bases. 2 3 Difference in bases= 𝐴𝐵 − 𝐶 𝐷 = 15𝑥 − 7𝑥 Half the difference in bases = 2 Let𝐿 be the length of leg CD. 2 2 2 𝐿 = 𝐻 + ( 3 15𝑥 −7𝑥 2 ) 2 2 𝐿 = 𝐻 + ( 2 3 15𝑥 −7𝑥 2 ) 2 Substitute in H. 𝐿 = 3 2 (12 8𝑥 ) + ( Simplify further. 2 3 15𝑥 −7𝑥 2 3 15𝑥 −7𝑥 2 ) 2 2 𝐿 = 1152𝑥 + 2 3 2 (15𝑥 −7𝑥 ) 4 1 c. Use the formula 2 𝑏ℎ to solve for area Substitute calculated values into the formula. 3 1 • 24𝑥 2𝑥 Area of BHD = 2 • 7𝑥 Multiply the constants, then multiply the powers of x, then combine 1 2 4 1 2 Area of BHD =84 • 𝑥 • 2 • 𝑥 9 Area of BHD = 84 2𝑥2 27. Distance to library is given :7 8 Simplify7 8 to14 2 This means the distance from Vanessa’s house to the library is14 2 Time it takes to get to the library = 14 2 8 2 7 = 4 ℎ𝑜𝑢𝑟𝑠 7 14 Since it's a round trip, that's two ways so 4 ℎ𝑜𝑢𝑟𝑠• 2 = 4 ℎ𝑜𝑢𝑟𝑠𝑓𝑜𝑟𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑛𝑡𝑜𝑡𝑎𝑙 14 5 − 4 = 1.5 Vanessa can only study at the library for 1.5 hours. 28. The area of the running circle is given by25 5. 2 2 Given the formula A =π𝑟 , we plug our area inso that25 5 =π𝑟 2 2 Manipulate the equation to isolate𝑟 , so 𝑟 = Take the square root of both sides so,𝑟 = 25 5 π 25 5 π Plug the numerator into the denesting formula and apply square root toπ. 𝑟 = 4 5 5 π Since 30% of the sector is not covered by shade, we can solve for it with Area not covered by shade =0. 3 • 25 5 = 7 . 5 5 Solve forθ using the equation 2 1 Area of sector = 2 θ𝑟 1 Plug7. 5 5 into the equation so that7. 5 5 = 2 θ( 25 5 ) π Solve for θ 3π θ = 5 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 Arc length = rθ 3π 4 3π 5 Arc length: 4 5 5 Arc length = ( 5 )( π ) 4 =3 π 5 π 29. We want to find x such that: 20• x = 100 Divide both sides by 20 such that x = 100 20 Simplify 20 =2 5 Substitute2 5 in so x = 100 2 5 Cancel out 100 for 50 in the numerator and 2 for 1 in the denominator x = 50 5 Multiply top and bottom by 5 50 5 x = 5• 5 x =10 5 x= about 22.36 → 23 classes 2 1 6 3 4 3 30. Pot 1 - 164 = 162 = 16= 4. | Pot 2 - = 168 = 164 = 16 → add both together → 4 + 4 3 16 31. Question : 8 = 2 2 72 8 18+15 6 Rewrite expression as 144 2 18+15 6 Multiply top and bottom by the conjugate of (18 + 15 6) , which is (18 − 15 6) . 144 2 18+15 6 • (18−15 6) (18−15 6) Simplify. =− 144 2−240 3 57 = − 2592 2− 4320 3 1026
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